Closed-Form Gaussian Information Projection for Embedded Simplices and Simplicial Mixtures

We study Gaussian approximation of Dirichlet--Gaussian probability density functions on embedded simplices. We derive the unique information projection onto the family of nondegenerate ambient Gaussian probability density functions, obtaining closed-form expressions for the optimal Gaussian and the minimum divergence for arbitrary positive Dirichlet parameters. The minimum divergence depends only on the intrinsic simplex dimension and Dirichlet parameters, is invariant up to an affine isomorphism, independent of the ambient dimension, and independent of the normal covariance. Furthermore, by joint convexity of Kullback--Leibler divergence, we establish explicit intrinsic dimension-dependent sharp upper bounds for finite Gaussian Mixture with shared component weights. For uniform probability density functions, these bounds reduce to dimension-dependent constants. The results provide closed-form Gaussian Mixture constructions and rigorous approximation guarantees for probability density functions associated with embedded simplicial complexes. A numerical example on a circular curve illustrates the theoretical results and their connections to geometric Gaussian Mixture representations.